Considering uncertainty in spatial models: causal inference, missing data imputations, and model comparison

Abstract

I present two applications of spatial and spatiotemporal models that stretch the traditional domain of these models, and a method for estimating the marginal likelihood in Bayesian models. Chapter 1 presents a spatial model for causal inference, in the setting of geographic regression discontinuity designs. The model is used to derive estimators of the treatment effect along a geographical border, to develop sensible estimands and estimators of the average treatment effect that account for the topology of the border, and to derive valid hypothesis tests. Chapter 2 addresses a bias in historical records of daily temperature minima and maxima due to the time of day at which the measurements are made, by deploying a spatiotemporal model in a missing data imputation framework. This is enabled by a novel and easy to implement Markov Chain Monte Carlo (MCMC) technique to sample the imputations conditionally on the observed daily extrema. Chapter 3 proposes a method for unbiasedly estimating the log ratio of normalizing constants of two unnormalized distributions, which can be used to estimate the marginal log likelihood of Bayesian models and for Bayesian cross-validation. These quantities are generally challenging to approximate from samples of the posterior, but are important for Bayesian model comparison and selection. I demonstrate the method on a spatial log Gaussian Cox process model and on a high-dimensional Bayesian variable selection model, and discuss how the lack of bias unlocks parallelization on multiple computing units.